University of Notre Dame
Aerospace and Mechanical Engineering

ME 654: Geometric Nonlinear Control
Homework 6

B. Goodwine Fall, 2003

Issued: August 28, 2003 Due: October 9, 2003

1. Investigate the bifurcations of equilibria for the system
   
   $$\dot x = f(x,\mu)$$
   
   
   near $\mu = 0$ with:
   1. $f(x,\mu) = \mu - x^2$
   2. $f(x,\mu) = \mu x - x^2$
   3. $f(x,\mu) = \mu^2 x - x^3$
   4. $f(x,\mu) = \mu + x^3$
   5. $f(x,\mu) = \mu^2 \alpha x + 2 \mu x^3 - x^5$ for various $\alpha$.
   In each case
   1. Use the conditions on $f(x,\mu)$ derived in class or on the handout to try to determine the type of bifurcation.
   2. Construct the bifurcation diagram.
2. Work through the calculus for the step in the derivation of conditions for the existence of a saddle-node bifurcation indicated in class.

Amicus Plato amicus Aristoteles magis amica veritas. [Plato is my friend, Aristotle is my friend, but my best friend is truth.]

--- Isaac Newton, undergraduate notebook, 1664.

Last updated: October 5, 2003.
B. Goodwine (jgoodwin@nd.edu)